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A modelbased control design approach for linear freepiston engines

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Kigezi, Tom Nsabwa and Dunne, Julian (2017) A model-based control design approach for linear free-piston engines. Journal of Dynamic Systems, Measurement and Control, 139 (11). ISSN 0022-0434

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Journal of Dynamic Systems Measurement and Control

A MODEL-BASED CONTROL DESIGN APPROACH

FOR LINEAR FREE-PISTON ENGINES

T. N. Kigezi

Department of Engineering and Design

The School of Engineering and Informatics

The University of Sussex, Falmer, Brighton, BN1 9QT, UK.

[email protected]

J. F. Dunne1

Department of Engineering and Design

The School of Engineering and Informatics

The University of Sussex, Falmer, Brighton, BN1 9QT, UK.

[email protected]

1Corresponding author © 2017 ASME

mailto:[email protected]

mailto:[email protected]


Summary

A general design approach is presented for model-based control of piston position in a free-piston engine (FPE). The proposed approach controls either ‘bottom-dead-centre’ (BDC) or ‘top-dead-centre’ (TDC) position. The key advantage of the approach is that it facilitates controller parameter selection, by way of deriving parameter combinations that yield both stable BDC and stable TDC. Driving the piston motion towards a target compression ratio is therefore achieved with sound engineering insight, consequently allowing repeatable engine cycles for steady power output. The adopted control design approach is based on linear control-oriented models derived from exploitation of energy conservation principles in a two-stroke engine cycle. Two controllers are developed: A Proportional Integral (PI) controller with an associated stability condition expressed in terms of controller parameters, and a Linear Quadratic Regulator (LQR) to demonstrate a framework for advanced control design where needed. A detailed analysis is undertaken on two FPE case studies differing only by rebound device type, reporting simulation results for both PI and LQR control. The applicability of the proposed methodology to other common FPE configurations is examined to demonstrate its generality.

26 main-section pages (double spaced) 27 references Figures 1 – 10 No Appendices


1. INTRODUCTION

Free-Piston engines (FPEs) are combustion-driven generators with controlled piston motion

that has none of the kinematic restrictions imposed by a slider-crank mechanism. In contrast,

piston motion in a conventional internal combustion (IC) engine is constrained by the fixed

stroke of a slider-crank mechanism. Free of such constraints, FPEs allow variable stroke and

compression ratio. Moreover, the absence of a crank mechanism means fewer moving parts,

with lower friction losses, and greater compactness. The output of an FPE is realized by

converting the piston force energy directly into electrical or hydraulic power. At present, electric

power generation is the most common application of FPEs targeted for deployment as Auxiliary

Power Units (APUs) in hybrid electric vehicles [1].

Because piston motion in an FPE is governed entirely by dynamic force interaction, active

piston motion control is needed for stable and repeatable engine cycles. As the piston dead-

centre positions (i.e. BDC or TDC) are free to vary cycle-by-cycle, accurate piston control is not

only essential to ensure sufficient scavenging and compression ratio for combustion, but also

ensures sufficient clearance to avoid the piston colliding with the cylinder-head. The central

problem then is control of compression ratio i.e. control of TDC or BDC position. Secondly,

controlling the piston to follow a given trajectory has been tested in starting the engine [2,3],

and in achieving robustness against misfire [4,5]. Additionally, cycle frequency manipulation is

viable in some applications and has been tested [6,7].

Although a number of prototype FPEs have been developed [8 - 10], no studies have adopted

a fully analytical model-based approach to piston motion control in a general way so as to

include various FPE configurations and types. Previous control approaches have largely

considered the engine as a ‘black box’ - the shortcoming being, no real justification for the

strategic basis adopted, and no corresponding stability assessment.


In TDC and BDC control, usually two separate control loops are involved. One control loop

achieves BDC control by regulating the fuel supply, whereas the second achieves TDC control

by regulating the energy stored in the rebound device, consequently regulating the rebound

‘stiffness’. Tikkanen and Vilenius [11] are early proponents of a similar approach, highlighting

the difficulty of achieving reliable piston motion control in practice. They proposed analytically-

guided control of TDC and BDC using total energy flows to control compression ratio via a

combination of fuel and piston load regulation – a potentially useful approach, although is left

untested. By contrast, Johansen et al [12,13] derive a detailed dynamic model of a diesel FPE.

Their control-oriented analysis reveals that TDC control can be achieved by varying rebound

stiffness whereas BDC control can be achieved by regulation of injected fuel per cycle. They

implement Proportional Integral (PI) and Proportional Integral Derivative (PID) controllers.

Similarly, Mikalsen and Roskilly [14,15] implement separate TDC and BDC control strategies

in their simulations of both spark ignition (SI) and compression ignition (CI) FPEs. They propose

TDC control by regulating rebound stiffness per cycle, and BDC control by fuel regulation.

Mikalsen and Roskilly [16] also identify the main difficulty of FPE control as being able to

achieve sufficient compression ratio across what they call the ‘entire load range’. This difficulty

is further addressed in [17], with PID and other approaches examined in [18].

This paper sets out to develop and achieve a general, model-based, analytical approach to

BDC and TDC control of a two-stroke FPE. In direct contrast to this work are non-model-based

attempts to control BDC and TDC, where engineering insight into the control problem is

achieved through trial and error – a potentially problematic approach, prone to unanticipated

engine responses. The proposed model-based approach has two important benefits:

(i) In controller parameter selection: A range of viable parameters to warrant stable BDC

and TDC can be computed prior to controller testing on hardware.

(ii) In availing a basis for advanced control design: A framework for advanced control


design is established, with the possibility of enforcing more stringent objectives other

than stability; for example, requirements on optimality, robustness or constraint

enforcement.

In the analysis presented here, energy conservation is exploited to derive control-oriented BDC

and TDC dynamic models. These models are subsequently used to obtain a formal FPE stability

condition in terms of the parameters of the widely-adopted PI controller. Furthermore, it is

demonstrated how the models can be used to develop advanced control strategies such as

Linear Quadratic Regulation (LQR) for optimality. This paper involves detailed extensions to

the work of Gong et al [19], where model-based control for TDC is developed for a specific FPE

configuration. A further step is taken in this paper to unify the approach into four common FPE

configuration cases.

The paper is structured as follows: Section 2 describes FPE modelling, Section 3 describes

control design and Section 4 considers four separate FPE configuration cases, generating

numerical results by simulation.

2. FREE PISTON ENGINE DYNAMIC MODELLING

This section broadly introduces two kinds of FPE model. First, a general piston dynamics and

gas thermodynamics model which captures fuel input and output power production – this will

be used for simulating the FPE. Second, BDC and TDC energy-based control-oriented models

are developed. These control-oriented models are used in BDC/TDC control design that follows

in Section 3. The scope of the modelling is now summarized by stating the following

assumptions:

(i) Zero-dimensional thermodynamic models are used to describe thermodynamic events

in the FPE. Whereas sufficient to demonstrate this paper’s BDC and TDC controller

effectiveness, these models are of limited scope to describe performance aspects such


as fuel efficiency or emissions formation.

(ii) All fuel available is assumed to be completely combusted, with negligible effects from

air-fuel ratio variability. Indeed, air-fuel ratio is regarded as a static parameter.

(iii) Ideal scavenging occurs, where all exhaust gas is completely expunged and

instantaneously replaced with fresh charge. Therefore, the effect of residual gases or

exhaust gas recirculation to combustion chamber thermodynamics is not considered.

(iv) As the focus is on achieving stable BDC and TDC control on the engine side,

investigation into the electrical energy conversion efficiency (and its variability with

BDC and TDC) on the generator side is deemed out of scope of this work.

The scope boundaries (i) - (iii) are not unusual in IC engine analysis for control design [20].

2.1 General Dynamic Engine Model

An idealised two-stroke FPE design concept is shown in Figure 1 comprising a single piston, a

translator rod, a permanent magnet generator, and a rebound device (which could either be a

mechanical spring or an air bounce chamber). Starting at BDC position bx , the piston is pushed

by the rebound device on the compression stroke to TDC position tx . Combustion takes place

in the trapped volume between tx and the cylinder head, driving the piston back to position bx

during the expansion stroke at the which scavenging occurs. Under ideal conditions, this cycle

repeats itself but in general bx and tx are free to vary from cycle-to-cycle to yield a variable

compression ratio. The electrical machine converts the piston rod thrust energy directly into a

useful output electrical power. In general, the output could also be hydraulic power.

To construct the equations of motion for a free piston engine, Newtonian mechanics and

combustion thermodynamics are used. Taking the compression stroke as the positive direction,

Newton’s 2nd law gives:

p G L RDm x F F F (1)


where pm is the piston-translator mass, and the forces on the right-hand-side are obtained as

follows: the in-cylinder gas force is given by G p GF A P , where GP is the in-cylinder gas pressure

which, other than at scavenging, can be obtained from the single-zone thermodynamics model

[21] as:

1G G ch ht

G

G G

dP dV dQ dQP

dt V dt V dt dt

(2)

where chQ is the gross heat release from fuel ignition, htQ is the heat transfer out of the

combustion chamber, GV is instantaneous cylinder volume, and is a heat capacity ratio of the

working gas. The gross heat release rate is given by:

chc LHV G

dxdQQ u

dt dt (3)

where LHVQ is the fuel lower heating value, Gu is the fuel mass, x is the fuel mass-fraction-

burned given by a time-based Wiebe function, and c is the combustion efficiency (usually 95%-

98% [21]), known to vary at least with air-fuel ratio [20]. The heat transfer rate is given by:

hts w

dQhA T T

dt (4)

where wT is the cylinder wall temperature, T is the gas temperature computed from the ideal

gas equation relating temperature, volume and pressure (from equation (2)), sA is the surface

area enclosing the combustion volume, and h is a heat transfer coefficient, for example given

by Hohenberg [22]. The FPE electrical power generation arises from a piston load force LF

assumed proportional to piston velocity i.e.

LF x (5)

where is an electrical machine constant typically called the generator coefficient. The

rebound device force depends on whether it is a spring or an air bounce chamber:


, mechanical spring

, bounce chambers

RDp RD

k xF

A P

(6)

where sk is the spring stiffness, and RDP corresponds to the bounce chamber pressure that in

general satisfies a polytropic process law: constantRD RDP V .

2.2 Control Oriented Engine Models

The general dynamic engine model above describes continuous behaviour of the piston

throughout a cycle. However, the piston only arrives at BDC or TDC once every cycle. The

control oriented models developed next, seek to capture this discrete cycle behaviour through

exploitation of energy conservation.

2.2.1 A ‘bottom dead centre’ (BDC) control-oriented energy balance model

A BDC control-oriented energy balance model is developed starting with simplifying

assumptions and definitions associated with the cycle of an FPE as follows:

i) A new cycle commences at the start of the compression stroke, at piston position bx .

ii) The end of the compression stroke occurs at piston position tx .

iii) The start of the expansion stroke occurs at piston position tx .

iv) The end of the expansion stroke occurs at piston positionb

x .

These simplifying assumptions and definitions (i) - (iv) allow two energy balance relations to be

constructed. The first is a compression-stroke energy balance statement as follows:

RD Gb tb t b t

W W E (7)

where RDb t

W

is work done by the rebound device in pushing the piston from bx to tx , Gb t

W

is work

done on the gas contained in cylinder when piston moves from bx to tx , and b tE is the energy

converted by piston motion associated with the useful output energy and friction, when the

piston moves from bx to tx . The second energy balance relation is an expansion-stroke

statement:


G RDt bt bt b

W W E (8)

where Gt b

W is the work done by the gas in the cylinder driving the piston from tx to

bx , RD

t b

W

is

the work done on the rebound device when the piston moves from tx tob

x , andt bE is the

energy converted by the piston motion associated with useful output energy and friction when

the piston moves from tx to b

x .

Addition of the energy balance equation (7) and equation (8), gives the full-cycle energy

balance:

0RD RD G Gb t t bb t b tt b t b

W W W W E E (9)

Equation (9) can be used to predict b

x when tx and bx are known. A visual description of the

piston motion in one cycle is given in Figure 2 (it shows the motion from one trough to the next,

as a solid line). Index 1,2,3,...k is used to denote the cycle count of the piston end points,

where Tx and Bx are the nominal end points, Gu is the fuel added to the cylinder for a given

cycle. The input variable RDu varies per cycle to adjust the stiffness of the rebound device. In

a bounce chamber for example, RDu is the trapped air mass which when varied per cycle

adjusts the bounce chamber's stiffness on a cycle-by-cycle basis. Variable RDu has no

relevance for a mechanical spring.

In general, assuming isentropic compression and expansion processes, the first two

parenthesized terms of equation (9) can be expressed in terms of the piston endpoint variables

1 1, , ,

k k k kb t b tx x x x and the input variables ,k kG RDu u - this generality is later demonstrated in the

development of three numerical case studies in Section 4 using the same six variables.

Furthermore, the last parenthesized term of equation (9) which represents the total energy


converted in a cycle, is assumed to be approximated by a polynomial function of the piston end

points, where, for small load changes, the total energy converted is approximately constant.

Equation (9) may alternatively be expressed as an implicit nonlinear function in the form:

1 11 , , , , , 0

k k k k k kb t b t G RDf x x x x u u (10)

By defining a nominal point for the variables of interest i.e.:

),,,( RDGTB UUxx (11)

where GU and RDU respectively depict the cylinder and rebound device inputs required to send

the piston from Tx to Bx and Bx to Tx , the associated error variables from nominal are

defined as:

b b B

t t T

G G G

RD RD RD

x x x

x x x

u u U

u u U

(12)

Equation (10), when expanded in the form of a Taylor series about the point , can be used to

generate the following predictive equation:

1 1 1 1k k k kb b Gx a x b u (13)

where:

11 1 1 1 1,k k k kt t RDc x d x e u g (14)

1111

111

111

111

111

111

111

111

,,

,,,

kkkkk

kkkkkk

bbRDbt

btbGbb

x

ffg

x

f

u

fe

x

f

x

fd

x

f

x

fc

x

f

u

fb

x

f

x

fa

(15a-f)

Equation (13) is the BDC control-oriented prediction model which describes the deviation from

nominal as a discrete-time, first order, linear time invariant (LTI) equation with input Gu , output


bx and known residual term 1 from the Taylor series expansion procedure, evaluated

according to equation (14).

2.2.2 A ‘top dead centre’ (TDC) control-oriented energy balance model

The development of a TDC control-oriented energy balance model is similar to the BDC control-

oriented model but with different end points, namely:

i) A new cycle starts at the beginning of the expansion stroke at piston position tx .

ii) The end of the expansion stroke occurs at piston position bx .

iii) The start of the compression stroke occurs at piston position bx .

iv) The end of the compression stroke occurs at piston positiont

x .

The corresponding expansion and compression stroke energy balance equation, analogous to

equation (9), is:

0RD RD G Gt b b tt b t bb t b t

W W W W E E (16)

Equation (16) can be used to predict t

x when bx and tx are known. Visualization making use

of Figure 3 helps to picture a full engine cycle (peak-to-peak on the solid line) and associated

piston end points, where k is the count index. Assuming isentropic compression and

expansion, the first two parenthesized terms of equation (16) can be directly expressed in terms

of the piston endpoint variables 1 1, , ,

k k k kt b t bx x x x as well as the input variables 1

, ,k k kRD G RDu u u .

The last parenthesized term in equation (16) can be expressed as a polynomial function of the

piston endpoint variables, which is nearly constant for small load changes. Similarly, equation

(16) may be equivalently expressed as an implicit function:

1 1 12 , , , , , , 0

k k k k k k kt b t b RD G RDf x x x x u u u (17)

which, when expanded as a Taylor series about and rearranged, yields the predictive

equation:


1 2 2 2k k k kt t RDx a x b u (18)

where:

1 12 2 2 2 2 2 ,

k k k k kb b G RDc x d x e u g u h (19)

and where:

1111

11111

222

222

222

222

222

222

222

, ,

, , , ,

kkkkk

kkkkkkkk

ttRDtG

tbtbtRDtt

x

ffh

x

f

u

fg

x

f

u

fe

x

f

x

fd

x

f

x

fc

x

f

u

fb

x

f

x

fa

(20a-g)

Equation (18) is the TDC control-oriented energy balance model which describes deviation from

nominal as a discrete time, first order, LTI equation with input RDu , output tx , and known

residual term 2 from the Taylor series expansion procedure, evaluated according to equation

(19).

3. CONTROL DESIGN

With BDC and TDC models constructed, design of feedback control action is possible to satisfy

specific control objectives. Equations (13) and (18) (when suffixes 1, 2, and t are omitted) are

of the same general form:

1k k k kx a x b u (21)

and can therefore be treated similarly in subsequent analysis. The control objective is to design

control action u to stabilize the output, i.e. to drive x to zero as k . But for convenience,

equation (21) can be simplified further, and by so doing, allows simplification of the subsequent

analysis. Consider an equivalent input v defined as:

k k kv b u (22)

which allows equation (21) to be rewritten as:


1k k kx a x v (23)

The control objective now becomes the design of an equivalent input v that drives the output

1kx to zero as k . Note that when the equivalent control v is designed for equation (23),

it is ultimately implemented for equation (21) as:

1k k ku v

b (24)

as per the relation between v and u in equation (22).

3.1 Proportional-Integral Control Design

Proportional-Integral (PI) control, which is well-suited to low order linear systems, has been

shown to provide effective control in simulation and experimental work on FPEs [3 – 6]. Since

equation (23) represents a first order linear system, PI control is appropriate for BDC and TDC

control of FPEs. Moreover, an associated stability condition can be derived. The input-output

transfer function for equation (23) is found as:

1

11

zG z

az

(25)

where z is the unit delay operator. For a reference value 0r , the feedback error is defined

as k ke r x . Defining the integral of the feedback error as 11 kkk IeI , a PI controller is

realised as:

k p k i kv k e k I (26)

where 0pk and 0ik . The transfer function from equation (26) (i.e. the feedback error to

control input) is given as:

1

1( )

1p i pk k k z

C zz

(27)

The plant model equation (25) and the controller (27) are in a closed negative-feedback loop,

whose transfer function relates the reference input to the output, and has the well-known form:


CG

CGzGCL

1)( (28)

with two poles 1p and 2p are evaluated as:

2 2

1 2

4 4,

2 2p p

(29)

where 1 , ( 1)pk a , and ( )i pk k a , in terms of the parameters pk and ik . For

stability of this closed loop system, the poles 1p and 2p must lie within the unit circle to imply

that the output x decays to zero as time goes to infinity. Another way to express this statement

is:

1 2 1 2 1p p p p (30)

which manifests a closed form stability condition for PI control in terms of parameters pk and

ik . In simple terms, for a given value of pk , the integral gain ik must be chosen to satisfy (30).

Indeed, a useful map showing regions of stable and unstable parameter combinations can

easily be generated, as is shown in Figure 4.

3.2 Advanced Control Design

Requirements, other than stability and output decay to zero, can be imposed on the control

action such as optimality, robustness, and even constraint enforcement. Here optimality of the

control action (relating to minimization of a mathematically defined performance index) is

considered i.e. achieving an optimal fuel supply or optimal regulation of the rebound device

stiffness. Linear quadratic regulation via a state space control design formalism is pursued for

illustrative purposes. For improved controller performance, integral action is applied to the

output as:

1k k kx (31)


By constructing the state vector Tw x , both equation (23) and (31) can then be directly

expressed in state space form as:

1k k kw Aw Bv (32)

where

0 1

,1 1 0

aA B

(33a, b)

To ensure stability of (32), and decay of w to zero as k , the control law is:

k kv Kw (34)

where the state feedback gain K is chosen to ensure that the eigenvalues of matrix A BK lie

within the unit circle. However, the optimal gain K that minimizes the performance objective

function:

0

( )T Tk k k k

k

J w Qw u Ru

(35)

is computed from:

1T TK B PB R B PA (36)

where Q and R in equation (35) are appropriately chosen positive-definite weight matrices, and

where P is a positive definite matrix that is a solution to the Riccati equation [23]:

10T T T TQ A PA P A PB B PB R B PA

(37)

4. CASE STUDIES - TESTING BY SIMULATION

The generic modelling and control design developed in the previous sections will now be

tailored to specific FPE configuration cases – all physically dissimilar, but with conceptually

identical configurations. Detailed development for each particular case will precede the test

simulation results. Two cases of FPEs, differing only by rebound device type, namely the case


of a mechanical spring, and the case of a bounce chamber, are first studied. In these studies,

Figures 2 and 3 shall be used for reference purposes. Examination will then follow for two other

common FPE cases, with one involving two opposed pistons in the same cylinder.

Note that in all simulations, the cylinder pressure is modelled using the Single Zone

thermodynamic model with a time-dependent Wiebe function for heat release adapted from

[21]. Perfect scavenging is assumed, with the intake pressure taken as standard atmospheric

pressure. The dynamic behavior of an FPE follows from [5, 8] where the electrical generator

force is assumed to be proportional to piston speed - a typical assumption with free-piston

engine generators (FPEGs). Table 1 gives the FPE geometric parameters used in simulations.

4.1 Case I: Mechanical Spring as Rebound Device

In this configuration, the FPE rebound device (label (5) in Figure 1) is simply a mechanical

spring [24] [25]. As the spring stiffness is fixed, the only control variable available for BDC

control is fuel supply. Whereas the objective of BDC control is to ensure the piston is driven to

nominal BDC, it is possible to compute the spring stiffness needed to send the piston from

nominal BDC to nominal TDC.

4.1.1 Detailed development of the control-oriented model for BDC control

The first task is to construct the specific form of equation (9). If the spring stiffness is denoted

by sk , the first parenthesized term in equation (9) becomes:

1

2 2 2 21 1

2 2k k k kRD RD s b t s t bb t t b

W W k x x k x x (38)

which, as expected, is a function only of the piston endpoints. Pressure varies with volume in

an isentropic process according to:

constant 1; compression stroke

constant 2; expansion strokePV (39)


where is the heat capacity ratio of the working gas. Therefore, from the isentropic work done,

the second parenthesized term is:

11

bbtttttbb GGGGGGGGG

tbG

btG

VPVPVPPVPWW (40)

where GP is in general the in-cylinder gas pressure, GP is the pressure rise at constant volume,

GV is the cylinder volume, and bGP is the air intake pressure during scavenging (which must be

known). Using equation (39) in equation (40):

11

11

btbbtbttt GGGGGGGGG

tbG

btG

VVVPVVVPPWW (41)

where the pressuretGP can be evaluated from equation (39) as:

b

t b

t

G

G G

G

VP P

V

(42)

Turning to the combustion pressure rise termtGP , if the total amount of fuel burned at constant

volume tGV is Gu , then the corresponding pressure rise is given by:

1

t t

t

c LHVG G

v G

Q RP u

c V

(43)

where c is the combustion efficiency, tGu is the fuel mass input for a given cycle, LHVQ is the

fuel lower heating value, R is the specific gas constant, and vc is the specific heat capacity of

the gas at constant volume.

Thus, using equations (39), (42), and (43), equation (40) can be expressed as a function of

cylinder volume and fuel input only. Since the cylinder volume depends on piston position (see

example formulation in (47)), equation (40) therefore depends only on the piston end points


and the fuel mass input. The third term of equation (9) as stated earlier, can, under low load

changes, be approximated as a constant i.e.:

1 2b t t bE E E E (44)

where 1E and 2E are constants. The sum of equations (38), (40), and (44) combine to form a

nonlinear equation of the form:

11 , , , 0

k k k kb t b Gf x x x u (45)

as shown by equation (10). A Taylor series expansion of equation (45) will generate the

particular form of the BDC control-oriented model for equation (13) – the subsequent controller

follows directly from the steps described in equations (21) – (37).

4.1.2 Assignment of the spring stiffness

The energy required by the piston to move from nominal BDC to nominal TDC during the

compression stroke is supplied entirely by the rebound device – in this case, a mechanical

spring. An appropriate choice of the spring stiffness ensures that the piston moving from

nominal BDC precisely reaches nominal TDC. This is achieved using the compression stroke

energy balance equation (7) evaluated at nominal piston end points. The spring stiffness

obtained is:

1 1

1

2 2

2 1 2 1

1b b tG G G c

s

B T

P V V r Ek

x x

(46)

where b tc G Gr V V is the nominal compression ratio, Bx and Tx are the nominal BDC and

nominal TDC piston displacement positions respectively.

4.1.3 TDC Estimation

Implementation of the control action in equation (24) for BDC control requires BDC feedback

(as kbx in kv ) and feedback of the immediately-following TDC position (as

ktx in k ). Whereas


the BDC feedback can be made available by a sensor, the immediately-following TDC position

must be estimated when the piston is at the BDC position. This can be done as follows: in

Figure 1, let l be the length from the left end of the cylinder to centre line 0x . Considering

the direction to the left of centre line 0x as positive, and to the right of centre line as negative,

then the instantaneous in-cylinder volume for a piston crown of area pA is given as:

G pV A l x (47)

Hence using equation (47), a TDC estimate is given as:

ˆ

ˆ tG

t

p

Vx l

A (48)

where tGV̂ is an estimate of the cylinder volume at the estimated TDC position tx̂ .

Using the compression stroke energy balance equation (7) and equation (48), an algebraic

equation can be constructed for ˆtGV as follows:

2 1ˆ ˆ ˆ 0t t tG G GpV qV rV s (49)

where the coefficients are:

2

2 21 2

11

2

11

11 ( )

2

b b

b b

sp

sp

G G

G G s b

p kA

q k lA

r P V

s P V E E k l x

(50a-d)

Equation (49) can be solved numerically, for example via Newton’s method, and used in

equation (48) to compute the TDC estimate. The iterations can be expected to converge quickly

given that an initial solution guess (for example nominal TDC) is not far from the true TDC


solution in a transient. In the simulation results, the number of iterations to find a solution was

never greater than 5.

To make the spring constant computation via equation (46) exact, the electrical generator can

be turned-off during the compression stroke, therefore rendering 1E equal to zero. The nominal

piston endpoints Bx and Tx are known, or easily calculated from the required compression

ratio. The nominal inputs Gu and RDu must be estimated – and the more accurate the estimates,

the better the controller performance.

4.1.4 Simulation Results and Discussion

Testing the control of BDC and TDC for the case of a mechanical spring as a rebound device

can now proceed. The FPE geometry is taken from Table 1, with the PI controller parameters

pk and ik selected from the stability map in Figure 4, and the LQR weighting parameters Q and

R selected as positive definite. It should be emphasized that only model-based control, such

as developed, allows the confident selection of the controller parameters i.e. from a pre-

determined set. The alternative is non-model-based control, which relies on a trial and error

approach to obtain meaningful engineering insight.

Figure 5 shows the piston error at BDC and TDC for both PI and LQR control, having started

with an offset and going to zero after a relatively small number of cycles. Hence a steady

compression ratio is achieved. The piston error at BDC and TDC is expressed as the

percentage:

Deviation from nominal BDC/TDC

100Nominal BDC/TDC

(51)

which must stay below a critical value (which for the geometry considered is 24%, indicating

where the deviation corresponds to the cylinder clearance length). In this case, the LQR control

transient is slower than the PI control transient, owing to a minimization of an objective function


that involves the fuel input (see equation (35)). Correspondingly the ‘Supplied fuel input’ in

Figure 5, shows that the LQR transient fuel supply is lower than that with PI control. However,

the FPE being an energy balance system at oscillations of constant amplitude (i.e. constant

compression ratio), the fuel supplied at steady state is the same amount required to overcome

a given load, regardless of the controller implemented. Therefore, the choice of one controller

over another should be made based on transient response performance.

The performance responses of the electrical power produced (deduced from equation (5)),

and piston oscillation frequency (or engine speed), are shown in Figure 6. As expected, when

steady compression ratio is achieved, both show stable convergence to the same value at

steady state. In the simulation, an initial piston position is chosen such that the initial BDC/TDC

error is small but significant (around 5%). In practice, a starting arrangement is required to bring

the piston from its rest position as close to nominal BDC/TDC as possible before engine firing.

This ensures that nominal compression ratio is achieved first. The largest possible initial

BDC/TDC error that yields a compression ratio sufficient for combustion can be investigated

experimentally.

4.2 Case II: Bounce Chamber as Rebound Device

In this configuration, the rebound device is a stiffness adjustable air bounce chamber (or gas

spring) [5, 6, 11, 12]. The chamber usually changes the air mass once every cycle to achieve

TDC control. By varying the air mass, the bounce chamber stiffness is varied.

4.2.1 Detailed development of the control-oriented model for BDC control

As in the previous example, the first task is to construct the BDC control model via equation

(9). Considering isentropic expansion and compression of the rebound device, this specializes

to:


11

111tbbbbbtbb

RDRDRDRDRDRDRDRDRD

btRD

tbRD

VVVVPVVVPWW (52)

Assuming an ideal gas, and denoting the mass of air in the bounce chamber as RDu , the

pressure term bRDP in equation (52) is evaluated according to the ideal gas law as:

1

b b b

b

RD RD RD

RD

P RT uV

(53)

where R is the specific gas constant, and bRDT is the air temperature at BDC (assumed to be a

known constant). The second parenthesized term of equation (9) remains as evaluated in

equations (41) – (43) because the cylinder side is no different from the previous example. Also,

the same approximation equation (44) holds. Thus, equation (9) is again expressed in general

nonlinear form:

1 11 , , , , , 0


Subsequent linearization by Taylor series expansion to achieve the BDC control-oriented model

equation (13), and subsequent control design is as described in equations (21) – (37).

4.2.2 Detailed development of the control-oriented model for TDC control

For the TDC model, the first parenthesized term of equation (16) can be adapted to the form:

11

11

tbttbtbb RDRDRDRDRDRDRDRD

btRD

tbRD

VVVPVVVPWW (55)

where the pressure termbRDP is computed as in equation (53). The pressure

tRDP is obtained from:

b

t b

t

RD

RD RD

RD

VP P

V

(56)

and where, according to the ideal gas law:


1

b b b

b

RD RD RD

RD

P RT uV

(57)

Using the same argument as in the previous example, the second parenthesized term of

equation (16) is:

11

11btbbtbttt

GGGGGGGGG

tbG

btG

VVVPVVVPPWW (58)

And similar to equation (42), the following condition holds:

b

t b

t

G

G G

G

VP P

V

(59)

where bGP is the air intake pressure during scavenging, and the pressure rise

tGP is as stated

in equation (43). The third parenthesized term of (16) is the same as that of equation (9) and

is therefore evaluated no differently from equation (44). Equation (16) can thus be stated in the

general nonlinear form:

1 1 12 , , , , , , 0

k k k k k k kt b t b RD G RDf x x x x u u u (60)

Taylor series expansion of equation (60) yields the control-oriented model corresponding to

equation (13) - subsequent controller design follows the process described by equations (21) –

(37).

4.2.3 TDC Estimation

Estimation of TDC is important in the implementation of the BDC controller. As in the previous

Case, the compression stroke energy balance is used to obtain:

1 1 1 11

ˆ ˆ 1 0b b t b b b t bRD RD RD RD G G G GP V V V P V V V E (61)


where ˆtRDV and ˆ

tGV are estimates of the rebound device volume and the cylinder volume at the

immediately-following TDC position respectively. Volumes ˆtRDV and ˆ

tGV are known functions of

ˆtx , thus when substituted in equation (61), ˆtx can be found as a direct solution.

4.2.4 Simulation Results and Discussion

The basic engine geometry for the numerical simulation is again given in Table 1, with the PI

controller parameters pk and ik selected from the stability map in Figure 4 and the LQR

weighting parameters Q and R selected as positive definite.

Figure 7 shows the piston response for PI and LQR control, plus the fuel supply input for both

controllers. Both TDC and BDC errors can be seen to converge to zero (implying convergence

to a steady compression ratio). Owing to minimization of the supplied input fuel in equation (35),

the LQR response transient is evidently slower than the PI response. But the LQR response

transient appears to be ‘smoother’, and on this basis, is preferable to the PI transient for the

parameters chosen. Note that at steady state, the same fuel is supplied regardless of the type

of controller.

The generated electrical power and the engine speed are shown in Figure 8, both converging

to their respective steady-state values when a steady compression ratio is achieved. There is

a brief initial deviation from a converging path for both transients. This can be attributed to the

interaction of the BDC and TDC controllers, as well as possibly unmodelled dynamics in control

design – for example, instantaneous fuel combustion is assumed during control design (see

equation (43)), whereas the engine is simulated with finite-time fuel combustion (see equation

(3)).

4.3 Case III: Combustion Chamber as Rebound Device

In this configuration (also known as a dual-piston FPE) the rebound device is a combustion

chamber [26-27] identical to the left-hand cylinder in Figure 1. The engine therefore comprises


two pistons on either end (hence the ‘dual-piston’ reference) which produces two power strokes

in a cycle – one in gas compression, and the other in gas expansion. The treatment of this case

reduces to analyzing two identical combustion chambers, but considering one as a rebound

device. Since a combustion chamber has already been accounted for in the previous two cases,

this third Case does not present any particular new challenge. The first parenthesized term of

equation (9) is found as:

11

11tbttbtbbb

RDRDRDRDRDRDRDRDRD

btRD

tbRD

VVVPVVVPPWW (62)

where tRDP is the rebound device intake pressure during scavenging. Following the arguments

used to obtain equations (42) and (43), bRDP and

bRDP are functions of: i) the rebound device

cylinder volume (which itself is expressible as a function of piston endpoints), and ii) the

rebound device fuel input RDu . Since the left-hand cylinder remains unchanged following-on

from the previous case, the second parenthesized term of equation (9) is evaluated in the same

way as in equation (41) – (43). Also, the same approximation equation (44) holds. This allows

equation (9) to be expressed in the general nonlinear form:

1 11 , , , , , 0


By Taylor series expansion, the control-oriented model equation (18) and subsequent controller

design, again follow from the procedure described in equations (21) – (37). By swapping the

cylinder functions on either end, the TDC control-oriented model is realized through the same

process as the BDC control-oriented model.

Figure 9 shows the BDC and TDC error responses using the same simulation settings as

described in the previous cases. As expected both errors converge to zero to yield a steady

compression ratio at steady state. The LQR response transient is slower – owing to a fuel


minimization requirement in (35) – but also less oscillatory than the PI controller response, for

the controller parameters used.

4.4 Case IV: Opposed Piston FPE

In this configuration, two opposing pistons share a combustion chamber to form an opposed

piston FPE [10] as shown in Figure 10. It is shown here that under symmetry conditions,

analysis of this configuration case for BDC and TDC control is no different from that for the

previously studied cases. Symmetry about the centre line simplifies analysis of the device, by

reducing the device to an equivalent single piston FPE configuration. This is achieved after

noting that on Figure 10, the common combustion volume is given by:

y xG G GV V V (64)

where yGV and

xGV are instantaneous gas chamber volumes on either side of the centre line.

Assuming symmetry of piston motion, and identical physical properties on either side of the

centre line, the two gas volumes are then equal i.e. y xG GV V giving:

2 2y xG G GV V V (65)

From equation (65), the common volume is equivalently-described either by the left or right gas

chamber volume. The common volume GV is the only form of coupling between the two pistons,

therefore symmetry acts as a decoupling condition. It can therefore be concluded from equation

(65) that opposed piston FPE analysis under symmetry conditions is equivalent to single piston

FPE analysis, but with twice the volume to the centre line. If the symmetry assumption does

not hold, then this amounts to quantifying the asymmetry between the two opposing piston FPE.

Knowledge of this asymmetry can then be compensated for in the equivalent single piston

model. Thus, under asymmetry conditions, by defining the volume as:

( )y xG GV V t (66)


where ( )t is the instantaneous time-varying volume difference between the left and right

cylinder volumes to the centre line, for which ( ) 0t implies complete symmetry. Substituting

equation (66) into equation (64) yields

2 ( )xG GV V t (67)

or, in terms of the left cylinder:

2 ( )yG GV V t (68)

Equations (67) and (68) are generalized forms of equation (65), taking into account asymmetry

of the left and right cylinders as quantified by parameter ( )t . The common volume GV is

described only in terms of the left or right cylinder volume to the centre line. The general finding

is that analysis of an opposed piston FPE configuration is equivalent to the analysis of just one

piston, for example in Cases I, II and III, assuming the level of asymmetry between the two

pistons is known, and adequately compensated for. It can be investigated whether the

asymmetry parameter ( )t can be modelled with simple and convenient functions that can be

fitted to experimental data. This serves as a possible future line of investigation.

5. CONCLUSIONS

A model-based procedure for control of BDC and TDC in a free-piston engine has been

developed, thereby achieving analytically-guided compression ratio control. The limited scope

of zero-dimensional thermodynamic modelling does not permit a first-principled investigation

into performance aspects such as fuel efficiency or emissions formation. However, the basic

objective of analytically deriving controller parameter combinations that produce stable

BDC/TDC responses has been achieved and demonstrated with PI control. Additionally, using

LQR control, advanced control yielding transient responses that satisfy stated performance


objectives has been demonstrated. Of greater significance however is the unified context in

which four FPE configurations can be treated to demonstrate the generality of the proposed

approach.


References [1] M. R. Hanipah, R. Mikalsen and A. Roskilly, "Recent commercial free piston engine developments for automotive applications," Applied Thermal Engineering, vol. 75, pp. 493-503, 2015. [2] K. Li, A. Sadighi and Z. Sun, "Active Motion Control of a Hydraulic Free Piston Engine," IEEE/ASME Transactions on Mechatronics, vol. 19, no. 4, pp. 1148-1159, 2014. [3] K. Li, C. Zhang and Z. Sun, "Precise piston trajectory control for a free piston engine," Control Engineering Practice, vol. 34, pp. 30-38, 2015. [4] H. Kosaka, T. Akita, K. Moriya, S. Goto, Y. Hotta, T. Umeno and K. Nakakita, "Development of Free Piston Engine Linear Generator System Part 1 - Investigation of Fundamental Characteristics," SAE Technical Paper 2014-01-1203, 2014. [5] S. Goto, K. Moriya, H. Kosaka, T. Akita, Y. Hotta, T. Umeno and K. Nakakita, "Development of Free Piston Engine Linear Generator System Part 2 - Investigation of Control System for Generator," SAE Technical Paper 2014-01-1193, 2014. [6] P. A. J. Achten, J. P. J. Van den Oever, J. Potma and G. Vael, "Horsepower with brains: The design of the Chiron free piston engine," SAE Paper 2000–01–2545, 2000. [7] A. Hbi and T. Ito, "Fundamental test results of a hydraulic free piston internal combustion engine," Proceedings of Institute of Mechanical Engineering, no. 218, pp. 1149–1157, 2004. [8] R. Mikalsen and A. Roskilly, "A review of free-piston engine history and applications," Applied Thermal Engineering, no. 27, pp. 2339-2352, 2007. [9] C. Toth-Nagy and N. N. Clark, "The Linear Engine in 2004," SAE Technical Paper 2005-01-2140, 2005. [10] P. A. J. Achten, "A review of free piston engine concepts," SAE Paper 941776, 1994. [11] S. Tikkanen and M. Vilenius, "Hydraulic Free Piston Engine – Challenge for Control," Proceedings of the 1999 European Control Conference, pp. 2943-2948, 1999. [12] T. A. Johansen, O. Egeland, E. A. Johannessen and R. Kvamsdal, "Free-Piston Diesel Engine Dynamics and Control," Proceedings of the American Control Conference, pp. 4579-4584, 2001. [13] T. A. Johansen, O. Egeland, E. A. Johannessen and R. Kvamsdal, "Free-Piston Diesel Engine Timing and Control—Toward Electronic Cam- and Crankshaft," IEEE Transactions on Control Systems Technology, vol. 10, no. 2, pp. 177-190, 2002. [14] R. Mikalsen and A. P. Roskilly, "The design and simulation of a two-stroke free-piston compression ignition engine for electrical power generation," Applied Thermal Engineering, no. 28, pp. 589-600, 2008. [15] R. Mikalsen and A. Roskilly, "Performance simulation of a spark ignited free-piston engine generator," Applied Thermal Engineering , no. 28, pp. 1726-33, 2008.


[16] R. Mikalsen and A. P. Roskilly, "The control of a free-piston engine generator. Part 1: Fundamental analyses," Applied Energy, vol. 87, pp. 1273-1280, 2009. [17] R. Mikalsen and A. P. Roskilly, "The control of a free-piston engine generator. Part 2: Engine dynamics and piston motion control," Applied Energy, vol. 87, pp. 1281–1287, 2010. [18] B. Jia, R. Mikalsen, A. Smallbone, Z. Zuo and H. Feng, "Piston motion control of a free-piston engine generator: A new approach," Applied Energy, vol. 179, pp. 1166–1175, 2016. [19] X. Gong, K. Zaseck, I. Kolmanovsky and H. Chen, "Modeling and Predictive Control of Free Piston Engine Generator," Proceedings of the 2015 American Control Conference, pp. 4735-4740, 2015. [20] L. Eriksson and L. Nielsen, Modeling and Control of Engines and Drivelines, John Wiley & Sons, 2014. [21] J. B. Heywood, Internal Combustion Engine Fundamentals, McGraw-Hill, Inc., 1988. [22] G. Hohenberg, "Advanced Approaches for Heat Transfer Calculations," SAE Paper 790825, 1979. [23] B. D. O. Anderson and J. B. Moore, Optimal Control, Linear Quadratic Methods, Englewood Cliffs, New Jersey: Prentice-Hall, Inc., 1989. [24] J. F. Dunne, "Dynamic Modelling and Control of Semifree-Piston Motion in a Rotary Diesel Generator Concept," Journal of Dynamic Systems, Measurement, and Control, vol. 132, no. 5, pp. 051003/1-051003/12, 2010. [25] V. Gopalakrishnan, P. M. P Najt and R. P. Durrett, "US Patent: Free Piston Linear Alternator Utilizing Opposed Pistons with Spring Return," 2014. [26] B. Jia, Z. Zuo, G. Tian, H. Feng and A. P. Roskilly, "Development and validation of a free-piston engine generator numeric model," Energy Conversion and Management, vol. 91, pp. 333-341, 2015. [27] P. Van Blarigan, N. Paradiso and S. Goldsborough, "Homogeneous Charge Compression Ignition with a Free Piston: A New Approach to Ideal Otto Cycle Performance," SAE Technical Paper 982484, 1998.


Table Captions

Table 1. Parameter values for Free Piston Engine simulations

Figure Captions Figure 1. Generic FPE schematic. The piston, translator, and generator permanent magnet (for this illustration of an FPE generator) constitute the moving mass. The rebound device may be a mechanical spring, an air bounce chamber or another cylinder. Figure 2. Visualization of piston motion over time annotated with notation used in analysis. One complete cycle is from b to b . The lines Tx and Bx are the nominal piston endpoints. The arrows represent inputs to the engine as fuel addition or rebound device stiffness adjustment. Figure 3. Visualization of piston motion over time annotated with notation used in analysis. One complete cycle is from t to t . The lines Tx and Bx are the nominal piston endpoints. The arrows represent inputs to the engine as fuel addition or rebound device stiffness adjustment. Figure 4. Parameter combinations pk , ik and associated regions of stability or instability, where

1.05a in system (24). Figure 5. BDC/TDC error and input fuel response for PI and LQR controllers with a mechanical spring as rebound device. Controller parameters were chosen within their stability bounds. LQR response transient is slower than the PI response transient owing to a minimization of input objective.

Figure 6. Output power and engine speed responses with a mechanical spring as rebound device. The same power and speed are achieved at steady state regardless of controller type.

Figure 7. BDC/TDC error and input fuel response for PI and LQR controllers with a bounce chamber as rebound device. Controller parameters were chosen within their stability bounds. LQR response transient is slower than the PI response transient owing to a minimization of input objective.

Figure 8. Output power and engine speed responses with a bounce chamber as rebound device. The same power and speed are achieved at steady state regardless of controller type. Figure 9. BDC/TDC error response for PI and LQR controllers with a combustion chamber as rebound device. Controller parameters were chosen within their stability bounds. LQR response transient is slower than the PI response transient owing to a minimization of input objective. Figure 10. An opposed piston FPE. Two pistons sharing a combustion volume oppose each other about the centre line.


List of Tables

Table 1. Parameter values for Free Piston Engine simulations

Parameter Value Nominal cylinder compression ratio 10.44 Nominal Cylinder displacement 0.5 litres Bore 86 mm Nominal Stroke 86 mm Piston-translator mass 9.0 kg Piston load (generator) coefficient 418.5 kg/s Nominal compression ratio of bounce chamber 10


List of Figures

Figure 1. Generic FPE schematic. The piston, translator, and generator permanent magnet (for this illustration of an FPE generator) constitute the moving mass. The rebound device may be a mechanical spring, an air bounce chamber or another cylinder.

tx 0x bx

1 2

34

5(1) Cylinder(2) Piston (3) Translator rod(4) Piston load

(generator)(5) Rebound

device


Figure 2. Visualization of piston motion over time annotated with notation used in analysis. One complete cycle is from b to b . The lines Tx and Bx are the nominal piston endpoints. The arrows represent inputs to the engine as fuel addition or rebound device stiffness adjustment.

Tx

Bx

1ktx

kbx

ktx

1kbx

kGu

kRDu

1kGu

t

bb

t


Figure 3. Visualization of piston motion over time annotated with notation used in analysis.

One complete cycle is from t to t . The lines Tx and Bx are the nominal piston endpoints. The arrows represent inputs to the engine as fuel addition or rebound device stiffness adjustment.

kGu

Tx

Bxkbx

ktx

1ktx

1kbx

t t

kRDu1kRDu

bb


Figure 4. Parameter combinations pk , ik and associated regions of stability or instability, where

1.05a in system (24).

Stable Unstable Unstable


Figure 5. BDC/TDC error and input fuel response for PI and LQR controllers with a mechanical spring as rebound device. Controller parameters were chosen within their stability bounds. LQR response transient is slower than the PI response transient owing to a minimization of

input objective.


Figure 6. Output power and engine speed responses with a mechanical spring as rebound device. The same power and speed are achieved at steady state regardless of controller type.


Figure 7. BDC/TDC error and input fuel response for PI and LQR controllers with a bounce chamber as rebound device. Controller parameters were chosen within their stability bounds. LQR response transient is slower than the PI response transient owing to a minimization of

input objective.


Figure 8. Output power and engine speed responses with a bounce chamber as rebound device. The same power and speed are achieved at steady state regardless of controller type.


Figure 9. BDC/TDC error for PI and LQR controllers with a combustion chamber as rebound device. Controller parameters were chosen within their stability bounds. LQR response transient is slower than the PI response transient owing to a minimization of input objective.


Figure 10. An opposed piston FPE. Two pistons sharing a combustion volume oppose each other about the centre line.

0x

xGV

0y

yGV

centre line

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